Chemical Thermodynamics and Flame Temperatures

INTRODUCTION

The parameters essential for the evaluation of combustion systems are the equilibrium product temperature and composition. If all the heat evolved in the reaction is employed solely to raise the product temperature, this temperature is called the adiabatic fl ame temperature. Because of the importance of the temperature and gas composition in combustion considerations, it is appropriate to review those aspects of the fi eld of chemical thermodynamics that deal with these subjects.

HEATS OF REACTION AND FORMATION

All chemical reactions are accompanied by either an absorption or evolution of energy, which usually manifests itself as heat. It is possible to determine this amount of heat—and hence the temperature and product composition—from very basic principles. Spectroscopic data and statistical calculations permit one to determine the internal energy of a substance. The internal energy of a given substance is found to be dependent upon its temperature, pressure, and state and is independent of the means by which the state is attained. Likewise, the change in internal energy, ΔE , of a system that results from any physical change or chemical reaction depends only on the initial and fi nal state of the system. Regardless of whether the energy is evolved as heat, energy, or work, the total change in internal energy will be the same. If a fl ow reaction proceeds with negligible changes in kinetic energy and potential energy and involves no form of work beyond that required for the fl ow, the heat added is equal to the increase of enthalpy of the system

Q =ΔH

where Q is the heat added and H is the enthalpy. For a nonfl ow reaction proceeding at constant pressure, the heat added is also equal to the gain in enthalpy

Q=ΔH

and if heat evolved,

Q=- ΔH

Most thermochemical calculations are made for closed thermodynamic systems, and the stoichiometry is most conveniently represented in terms of the molar quantities as determined from statistical calculations. In dealing with compressible fl ow problems in which it is essential to work with open thermodynamic systems, it is best to employ mass quantities. Throughout this text uppercase symbols will be used for molar quantities and lowercase symbols for mass quantities.

One of the most important thermodynamic facts to know about a given chemical reaction is the change in energy or heat content associated with the reaction at some specified temperature, where each of the reactants and products is in an appropriate standard state. This change is known either as the energy or as the heat of reaction at the specified temperature.

The standard state means that for each state a reference state of the aggregate exists. For gases, the thermodynamic standard reference state is the ideal gaseous state at atmospheric pressure at each temperature. The ideal gaseous state is the case of isolated molecules, which give no interactions and obey the equation of state of a perfect gas. The standard reference state for pure liquids and solids at a given temperature is the real state of the substance at a pressure of 1 atm. As discussed before, understanding this definition of the standard reference state is very important when considering the case of high-temperature combustion in which the product composition contains a substantial mole fraction of a condensed phase, such as a metal oxide.

The thermodynamic symbol that represents the property of the substance in the standard state at a given temperature is written, for example, as Ht° , Et° , etc., where the “ degree sign ” superscript ° specifi es the standard state, and the subscript T the specifi c temperature. Statistical calculations actually permit the determination of Et - Eo , which is the energy content at a given temperature referred to the energy content at 0 K. For 1 mol in the ideal gaseous state,

(1.1)

(1.2)

which at 0 K reduces to

(1.3)

Thus the heat content at any temperature referred to the heat or energy content at 0 K is known and

(1.4)

The value (E°= E°o ) is determined from spectroscopic information and is actually the energy in the internal (rotational, vibrational, and electronic) and external (translational) degrees of freedom of the molecule. Enthalpy (H°- H°o) has meaning only when there is a group of molecules, a mole for instance; it is thus the Ability of a group of molecules with internal energy to do PV work. In this sense, then, a single molecule can have internal energy, but not enthalpy. As stated, the use of the lowercase symbol will signify values on a mass basis. Since flame temperatures are calculated for a closed thermodynamic system and molar conservation is not required, working on a molar basis is most convenient. In flame propagation or reacting flows through nozzles, conservation of mass is a requirement for a convenient solution; thus when these systems are considered, the per unit mass basis of the thermochemical properties is used. From the definition of the heat of reaction, Qp will depend on the temperature T at which the reaction and product enthalpies are evaluated. The heat of reaction at one temperature To can be related to that at another temperature T1 . Consider the reaction configuration shown in Fig. 1.1 . According to the First Law of Thermodynamics, the heat changes that proceed from reactants at temperature To to products at temperature T1, by either path A or path B must be the same. Path A raises the reactants from temperature To to T1, and reacts at T1. Path B reacts at To and raises the products from To to T1 . This energy equality, which relates the heats of reaction at the two different temperatures, is written as

(1.5)

where n specifies the number of moles of the ith product or jth reactant. Any phase changes can be included in the heat content terms. Thus, by knowing the difference in energy content at the different temperatures for the products and reactants, it is possible to determine the heat of reaction at one temperature from the heat of reaction at another.

Figure 1.1 H eats of reactions at different base temperatures.

If the heats of reaction at a given temperature are known for two separate reactions, the heat of reaction of a third reaction at the same temperature may be determined by simple algebraic addition. This statement is the Law of Heat Summation. For example, reactions (1.6) and (1.7) can be carried out conveniently in a calorimeter at constant pressure:

(1.6)

(1.7)

Subtracting these two reactions, one obtains

(1.8)

Since some of the carbon would burn to CO₂ and not solely to CO, it is difficult to determine calorimetrically the heat released by reaction (1.8).

It is, of course, not necessary to have an extensive list of heats of reaction to determine the heat absorbed or evolved in every possible chemical reaction. A more convenient and logical procedure is to list the standard heats of formation of chemical substances. The standard heat of formation is the enthalpy of a substance in its standard state referred to its elements in their standard states at the same temperature. From this definition it is obvious that heats of formation of the elements in their standard states are zero.

The value of the heat of formation of a given substance from its elements may be the result of the determination of the heat of one reaction. Thus, from the calorimetric reaction for burning carbon to CO₂ [Eq. (1.6)], it is possible to write the heat of formation of carbon dioxide at 298 K as

The superscript to the heat of formation symbol ∆H°_f represents the standard state, and the subscript number represents the base or reference temperature. From the example for the Law of Heat Summation, it is apparent that the heat of formation of carbon monoxide from Eq. (1.8) is

It is evident that, by judicious choice, the number of reactions that must be measured calorimetrically will be about the same as the number of substances whose heats of formation are to be determined.

The logical consequence of the preceding discussion is that, given the heats of formation of the substances comprising any particular reaction, one can directly determine the heat of reaction or heat evolved at the reference temperature T₀ , most generally T₂₉₈, as follows:

(1.9)

Extensive tables of standard heats of formation are available, but they are not all at the same reference temperature. The most convenient are the compilations known as the JANAF [1] and NBS Tables [2] , both of which use 298 K as the reference temperature. Table 1.1 lists some values of the heat of formation taken from the JANAF Thermochemical Tables. Actual JANAF tables are reproduced in Appendix A. These tables, which represent only a small selection from the JANAF volume, were chosen as those commonly used in combustion and to aid in solving the problem sets throughout this book. Note that, although the developments throughout this book take the reference state as 298 K, the JANAF tables also list ∆H°_f for all temperatures. When the products are measured at a temperature T₂ different from the reference temperature T₀ and the reactants enter the reaction system at a temperature T₀ different from the reference temperature, the heat of reaction becomes

(1.10)

The reactants in most systems are considered to enter at the standard reference temperature 298 K. Consequently, the enthalpy terms in the braces for the reactants disappear. The JANAF tables tabulate, as a putative convenience, (H°_t -H°₂₉₈)  instead of (H°_t -H°₀). This type of tabulation is unfortunate since the reactants for systems using cryogenic fuels and oxidizers, such as those used in rockets, can enter the system at temperatures lower than the reference temperature. Indeed, the fuel and oxidizer individually could enter at different temperatures. Thus the summation in Eq. (1.10) is handled most conveniently by realizing that  T’₀ may vary with the substance j.

The values of heats of formation reported in Table 1.1 are ordered so that the largest positive values of the heats of formation per mole are the highest and those with negative heats of formation are the lowest. In fact, this table is similar to a potential energy chart. As species at the top react to form species at the bottom, heat is released, and an exothermic system exists. Even a species that has a negative heat of formation can react to form products of still lower negative heats of formation species, thereby releasing heat. Since some fuels that have negative heats of formation form many moles of product species having negative heats of formation, the heat release in such cases can be large. 

Equation (1.9) shows this result clearly. Indeed, the first summation in Eq. (1.9) is generally much greater than the second. Thus the characteristic of the reacting species or the fuel that significantly determines the heat release is its chemical composition and not necessarily its molar heat of formation. As explained in Section D2, the heats of formation listed on a per unit mass basis simplifi es one’s ability to estimate relative heat release and temperature of one fuel to another without the detailed calculations reported later in this chapter and in Appendix I.

The radicals listed in Table 1.1 that form their respective elements have their heat release equivalent to the radical’s heat of formation. It is then apparent that this heat release is also the bond energy of the element formed. Non-radicals such as acetylene, benzene, and hydrazine can decompose to their elements and/ or other species with negative heats of formation and release heat. Consequently, these fuels can be considered rocket monopropellants. Indeed, the same would hold for hydrogen peroxide; however, what is interesting is that ethylene oxide has a negative heat of formation, but is an actual rocket monopropellant because it essentially decomposes exothermically into carbon monoxide and methane [3] .

Chemical reaction kinetics restricts benzene, which has a positive heat of formation from serving as a monopropellant because its energy release is not sufficient to continuously initiate decomposition in a volumetric reaction space such as a rocket combustion chamber. Insight into the fundamentals for understanding this point is covered in Chapter 2, Section B1. Indeed, for acetylene type and ethylene oxide monopropellants the decomposition process must be initiated with oxygen addition and spark ignition to then cause self-sustained decomposition. Hydrazine and hydrogen peroxide can be ignited and self-sustained with a catalyst in a relatively small volume combustion chamber. Hydrazine is used extensively for control systems, back pack rockets, and as a bipropellant fuel. It should be noted that in the Gordon and McBride equilibrium thermodynamic program [4] discussed in Appendix I, the actual results obtained might not be realistic because of kinetic reaction conditions that take place in the short stay times in rocket chambers. For example, in the case of hydrazine, ammonia is a product as well as hydrogen and nitrogen [5] . The overall heat release is greater than going strictly to its elements because ammonia is formed in the decomposition process and is frozen in its composition before exiting the chamber. Ammonia has a relatively large negative heat of formation.

Referring back to Eq. (1.10), when all the heat evolved is used to raise the temperature of the product gases, ΔH and Qp become zero. The product temperature T₂ in this case is called the adiabatic flame temperature and Eq. (1.10) becomes

(1.11)

Again, note that T’₀ can be different for each reactant. Since the heats of formation throughout this text will always be considered as those evaluated at the reference temperature T₀=298K, the expression in braces becomes {(H°_t - H°₀) – (H°_t0 - H°₀)} = (H°_t - H°_t0), which is the value listed in the JANAF tables (see Appendix A).

If the products ni of this reaction are known, Eq. (1.11) can be solved for the fl ame temperature. For a reacting lean system whose product temperature is less than 1250 K, the products are the normal stable species CO₂, H₂O, N₂, and O₂, whose molar quantities can be determined from simple mass balances. However, most combustion systems reach temperatures appreciably greater than 1250 K, and dissociation of the stable species occurs. Since the dissociation reactions are quite endothermic, a small percentage of dissociation can lower the fl ame temperature substantially. The stable products from a C-H- O reaction system can dissociate by any of the following reactions:

Each of these dissociation reactions also specifies a definite equilibrium concentration of each product at a given temperature; consequently, the reactions are written as equilibrium reactions. In the calculation of the heat of reaction of low-temperature combustion experiments the products could be specified from the chemical stoichiometry; but with dissociation, the specification of the product concentrations becomes much more complex and the n_i’s in the flame temperature equation [Eq. (1.11)] are as unknown as the flame temperature itself. In order to solve the equation for the n_i’s and T₂ , it is apparent that one needs more than mass balance equations. The necessary equations are found in the equilibrium relationships that exist among the product composition in the equilibrium system.

FREE ENERGY AND THE EQUILIBRIUM CONSTANTS

The condition for equilibrium is determined from the combined form of the first and second laws of thermodynamics; that is,

(1.12)

where S is the entropy. This condition applies to any change affecting a system of constant mass in the absence of gravitational, electrical, and surface forces. However, the energy content of the system can be changed by introducing more mass. Consider the contribution to the energy of the system on adding one molecule i to be μ_i. The introduction of a small number dni of the same type contributes a gain in energy of the system of μidni. All the possible reversible increases in the energy of the system due to each type of molecule i can be summed to give

(1.13)

It is apparent from the definition of enthalpy H and the introduction of the concept of the Gibbs free energy G

(1.14)

that

(1.15)

and

(1.16)

Recall that P and T are intensive properties that are independent of the size of mass of the system, whereas E, H, G, and S (as well as V and n ) are extensive properties that increase in proportion to mass or size. By writing the general relation for the total derivative of G with respect to the variables in Eq. (1.16), one obtains

(1.17)

Thus,

(1.18)

or, more generally, from dealing with the equations for E and H

(1.19)

where µ_i is called the chemical potential or the partial molar free energy. The condition of equilibrium is that the entropy of the system have a maximum value for all possible configurations that are consistent with constant energy and volume. If the entropy of any system at constant volume and energy is at its maximum value, the system is at equilibrium; therefore, in any change from its equilibrium state dS is zero. It follows then from Eq. (1.13) that the condition for equilibrium is